Documentation

Mathlib.CategoryTheory.Bicategory.Extension

Extensions and lifts in bicategories #

We introduce the concept of extensions and lifts within the bicategorical framework. These concepts are defined by commutative diagrams in the (1-)categorical context. Within the bicategorical framework, commutative diagrams are replaced by 2-morphisms. Depending on the orientation of the 2-morphisms, we define both left and right extensions (likewise for lifts). The use of left and right here is a common one in the theory of Kan extensions.

Implementation notes #

We define extensions and lifts as objects in certain comma categories (StructuredArrow for left, and CostructuredArrow for right). See the file CategoryTheory.StructuredArrow for properties about these categories. We introduce some intuitive aliases. For example, LeftExtension.extension is an alias for Comma.right.

References #

@[inline, reducible]
abbrev CategoryTheory.Bicategory.LeftExtension {B : Type u} [CategoryTheory.Bicategory B] {a : B} {b : B} {c : B} (f : a b) (g : a c) :
Type (max v w)

Triangle diagrams for (left) extensions.

  b
  △ \
  |   \ extension  △
f |     \          | unit
  |       ◿
  a - - - ▷ c
      g
Instances For
    @[inline, reducible]
    abbrev CategoryTheory.Bicategory.LeftExtension.extension {B : Type u} [CategoryTheory.Bicategory B] {a : B} {b : B} {c : B} {f : a b} {g : a c} (t : CategoryTheory.Bicategory.LeftExtension f g) :
    b c

    The extension of g along f.

    Instances For
      @[inline, reducible]

      The 2-morphism filling the triangle diagram.

      Instances For

        The left extension along the identity.

        Instances For
          @[inline, reducible]
          abbrev CategoryTheory.Bicategory.LeftLift {B : Type u} [CategoryTheory.Bicategory B] {a : B} {b : B} {c : B} (f : b a) (g : c a) :
          Type (max v w)

          Triangle diagrams for (left) lifts.

                      b
                    ◹ |
             lift /   |      △
                /     | f    | unit
              /       ▽
            c - - - ▷ a
                 g
          
          Instances For
            @[inline, reducible]
            abbrev CategoryTheory.Bicategory.LeftLift.lift {B : Type u} [CategoryTheory.Bicategory B] {a : B} {b : B} {c : B} {f : b a} {g : c a} (t : CategoryTheory.Bicategory.LeftLift f g) :
            c b

            The lift of g along f.

            Instances For
              @[inline, reducible]

              The 2-morphism filling the triangle diagram.

              Instances For

                The left lift along the identity.

                Instances For
                  @[inline, reducible]
                  abbrev CategoryTheory.Bicategory.RightExtension {B : Type u} [CategoryTheory.Bicategory B] {a : B} {b : B} {c : B} (f : a b) (g : a c) :
                  Type (max v w)

                  Triangle diagrams for (right) extensions.

                    b
                    △ \
                    |   \ extension  | counit
                  f |     \          ▽
                    |       ◿
                    a - - - ▷ c
                        g
                  
                  Instances For
                    @[inline, reducible]

                    The extension of g along f.

                    Instances For
                      @[inline, reducible]

                      The 2-morphism filling the triangle diagram.

                      Instances For

                        The right extension along the identity.

                        Instances For
                          @[inline, reducible]
                          abbrev CategoryTheory.Bicategory.RightLift {B : Type u} [CategoryTheory.Bicategory B] {a : B} {b : B} {c : B} (f : b a) (g : c a) :
                          Type (max v w)

                          Triangle diagrams for (right) lifts.

                                      b
                                    ◹ |
                             lift /   |      | counit
                                /     | f    ▽
                              /       ▽
                            c - - - ▷ a
                                 g
                          
                          Instances For
                            @[inline, reducible]
                            abbrev CategoryTheory.Bicategory.RightLift.lift {B : Type u} [CategoryTheory.Bicategory B] {a : B} {b : B} {c : B} {f : b a} {g : c a} (t : CategoryTheory.Bicategory.RightLift f g) :
                            c b

                            The lift of g along f.

                            Instances For
                              @[inline, reducible]

                              The 2-morphism filling the triangle diagram.

                              Instances For

                                The right lift along the identity.

                                Instances For